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Macedonian Mathematical Competitions

Algebra Difficulty 5.0 AIME Prove it North Macedonia

Mile imagined a number and said: "If I multiply the number by two and add this product to the half of the number I will get a number that is for 26 bigger than the third of the imagined number." What number did Mile imagine?

Solution

Let $x$ be the number that Mile imagined. Then we solve the equation $2x + \frac{1}{2}x = \frac{1}{3}x + 26$ , from where we get $12x + 3x = 2x + 156$ or $13x = 156$ , $x = 12$ . Mile imagined the number 12.

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